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Home , Ali Chamseddine, Electroweak epoch, Inflationary epoch

Early Models

Matilde Marcolli

Ma148b Spring 2016 Topics in Mathematical

Matilde Marcolli Early Universe Models This lecture based on Matilde Marcolli, Elena Pierpaoli, Early universe models from Noncommutative , Advances in Theoretical and , Vol.14 (2010) N.5, 1373–1432. Daniel Kolodrubetz, Matilde Marcolli, Boundary conditions of the RGE flow in the noncommutative geometry approach to and , Phys. Lett. B, Vol.693 (2010) 166–174 Re-examine RGE flow; gravitational terms in the asymptotic form of the spectral action; cosmological timeline; running in the very early universe; inflation and other gravitational phenomena

Matilde Marcolli Early Universe Models Asymptotic expansion of spectral action for large energies Z Z 1 √ 4 √ 4 S = 2 R g d x + γ0 g d x 2κ0 Z Z µνρσ√ 4 ∗ ∗√ 4 + α0 Cµνρσ C g d x + τ0 R R g d x 1 Z √ Z √ + |DH|2 g d 4x − µ2 |H|2 g d 4x 2 0 Z Z 2 √ 4 4 √ 4 − ξ0 R |H| g d x + λ0 |H| g d x 1 Z √ + (G i G µνi + F α F µνα + B Bµν ) g d 4x 4 µν µν µν

A modified model with non minimal coupling to Higgs and minimal coupling to gauge theories

Matilde Marcolli Early Universe Models Coefficients and parameters: 2 1 96f2Λ − f0c 1 4 2 f0 2κ2 = 2 γ0 = 2 (48f4Λ − f2Λ c + d) 0 24π π 4 3f 11f α = − 0 τ = 0 0 10π2 0 60π2 2 2 2 f2Λ e 1 π b µ0 = 2 − ξ0 = 12 λ0 = 2 f0 a 2f0a

f0, f2, f4 free parameters, f0 = f (0) and, for k > 0, Z ∞ k−1 fk = f (v)v dv. 0 a, b, c, d, e functions of Yukawa parameters of νMSM † † † † a = Tr(Yν Yν + Ye Ye + 3(Yu Yu + Yd Yd )) † 2 † 2 † 2 † 2 b = Tr((Yν Yν) + (Ye Ye ) + 3(Yu Yu) + 3(Yd Yd ) ) c = Tr(MM†) d = Tr((MM†)2)

† † e = Tr(MM Yν Yν).

Matilde Marcolli Early Universe Models Two different perspectives on running: parameters a, b, c, d, e run with RGE

The relation between coefficients κ0, γ0, α0, τ0, µ0, ξ0, λ0 and parameters a, b, c, d, e hold only at Λunif : constraint on initial conditions; independent running

There is a range of energies Λmin ≤ Λ ≤ Λunif where the running of coefficients κ0, γ0, α0, τ0, µ0, ξ0, λ0 determined by running of a, b, c, d, e and relation continues to hold (very early universe only) The first perspective requires independent running of gravitational parameters with given condition at unification; the second perspective allows for interesting gravitational phenomena in the very early universe specific to NCG model only

Matilde Marcolli Early Universe Models The first approach followed in Ali Chamseddine, , Matilde Marcolli, Gravity and the with mixing, ATMP 11 (2007) 991–1090, arXiv:hep-th/0610241 For modified gravity models Z  1 ω θ  √ C C µνρσ − R2 + R∗R∗ g d4x 2η µνρσ 3η η Running of gravitational parameters (Avramidi, Codello–Percacci, Donoghue) by 1 133 β = − η2 η (4π)2 10 1 25 + 1098 ω + 200 ω2 β = − η ω (4π)2 60 1 7(56 − 171 θ) β = η. θ (4π)2 90

Matilde Marcolli Early Universe Models With relations at unification get running like

log10 Μ GeV 5 10 15 log10 Μ GeV 4 6 8 10 12 14 16

"0.4 ! " # ! " # "0.005 " 0.6 Ω log Μ MZ

"0.8 "0.010 ! ! " ## Η log Μ MZ "1.0

! ! " ## "0.015 "1.2

log10 Μ GeV 5 10 15

! " # "0.2

Θ log Μ MZ "0.4

! ! " ##

"0.6 Plausible values in low energy limit (near fixed points) Look then at second point of view: M.M.-E.Pierpaoli (2009)

Matilde Marcolli Early Universe Models Renormalization group equations for SM with right handed and Majorana mass terms, from unification energy (2 × 1016 GeV) down to the (102 GeV) AKLRS S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024. Remark: RGE analysis in [CCM] only done using minimal SM

Matilde Marcolli Early Universe Models df 1-loop RGE equations: Λ dΛ = βf (Λ) 19 41 16π2 β = b g 3 with (b , b , b ) = (−7, − , ) gi i i SU(3) SU(2) U(1) 6 10 3 3 17 9 16π2 β = Y ( Y †Y − Y †Y + a − g 2 − g 2 − 8g 2) Yu u 2 u u 2 d d 20 1 4 2 3 3 3 1 9 16π2 β = Y ( Y †Y − Y †Y + a − g 2 − g 2 − 8g 2) Yd d 2 d d 2 u u 4 1 4 2 3 3 3 9 9 16π2 β = Y ( Y †Y − Y †Y + a − g 2 − g 2) Yν ν 2 ν ν 2 e e 20 1 4 2 3 3 9 9 16π2 β = Y ( Y †Y − Y †Y + a − g 2 − g 2) Ye e 2 e e 2 ν ν 4 1 4 2 2 † † T 16π βM = Yν Yν M + M(Yν Yν ) 3 3 3 16π2 β = 6λ2 − 3λ(3g 2 + g 2) + 3g 4 + ( g 2 + g 2)2 + 4λa − 8b λ 2 5 1 2 2 5 1 2 2 Note: different normalization from [CCM] and 5/3 factor included in g1

Matilde Marcolli Early Universe Models Method of AKLRS: non-degenerate spectrum of Majorana masses, different effective field theories in between the three see-saw scales:

RGE from unification Λunif down to first see-saw scale (largest eigenvalue of M) (3) Introduce Yν removing last row of Yν in basis where M diagonal and M(3) removing last row and column. Induced RGE down to second see-saw scale (2) (2) Introduce Yν and M , matching boundary conditions Induced RGE down to first see-saw scale (1) (1) Introduce Yν and M , matching boundary conditions

Induced RGE down to electoweak energy Λew (N) (N) Use effective field theories Yν and M between see-saw scales

Matilde Marcolli Early Universe Models Running of coefficients a, b with RGE

1.07 1.90

1.06 1.89

1.05

1.88 1.04

2´1014 4´1014 6´1014 8´1014 1´1015 2´1014 4´1014 6´1014 8´1014 1´1015 Coefficients a and b near the top see-saw scale

Matilde Marcolli Early Universe Models Running of coefficient c with RGE

3.45383´1029 3.49´1029 3.45392´1029

29 3.45390´1029 3.45383´10 3.48´1029

29 3.45388´10 3.45383´1029 3.47´1029 3.45386´1029

4.1´1012 4.2´1012 4.3´1012 4.4´1012 4.5´1012 2´1014 4´1014 6´1014 8´1014 1´1015 1.05´10141.10´10141.15´10141.20´10141.25´10141.30´1014 Running of coefficient d with RGE 1.10558´1059 59 1.10560´1059 1.10558´10 1.10558´1059 1.10560´1059 1.10558´1059 1.10558´1059 1.10560 1059 ´ 1.10558´1059 1.10558´1059 1.10559´1059 1.10558´1059 59 1.10558´1059 1.10558´10

1.05´10141.10´10141.15´10141.20´10141.25´10141.30´1014 1´1014 2´1014 3´1014 4´1014 5´1014 3.5´1012 4.0´1012 4.5´1012 5.0´1012 Effect of the three see-saw scales

Matilde Marcolli Early Universe Models Running of coefficient e with RGE 3.0´1029 6.215´1027 2.5´1029 6.210´1027 6.1865´1027 2.0´1029 6.205´1027

29 1.5´10 6.200´1027 6.1860´1027 29 1.0´10 6.195´1027

5.0´1028 6.190´1027 2´1012 4´1012 6´1012 8´1012 1´1013

2´1014 4´1014 6´1014 8´1014 1´1015 5.0´1013 1.0´1014 1.5´1014 2.0´1014 Effect of the three see-saw scales With default boundary conditions at unification of AKLRS ...but sensitive dependence on the initial conditions ⇒ fine tuning

Matilde Marcolli Early Universe Models Changing initial conditions: maximal mixing conditions at unification ζ = exp(2πi/3)  1 ζ ζ2  1 U (Λ ) = ζ 1 ζ PMNS unif 3   ζ2 ζ 1  12.2 × 10−9 0 0  1 δ = 0 170 × 10−6 0 (↑1) 246   0 0 15.5 × 10−3

† Yν = UPMNS δ(↑1)UPMNS

Daniel Kolodrubetz, Matilde Marcolli, Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology, arXiv:1006.4000

Matilde Marcolli Early Universe Models Effect on coefficients running

Matilde Marcolli Early Universe Models Matilde Marcolli Early Universe Models Further evidence of sensitive dependence: changing only one parameter in diagonal matrix Yν get running of top term:

Matilde Marcolli Early Universe Models Geometric constraints at unification energy λ parameter constraint

2 π b(Λunif ) λ(Λunif ) = 2 2f0 a(Λunif ) Higgs vacuum constraint √ af 2M 0 = W π g See-saw mechanism and c constraint

2 2 2f2Λunif 6f2Λunif ≤ c(Λunif ) ≤ f0 f0 Mass relation at unification

X 2 2 2 2 2 (mν + me + 3mu + 3md )|Λ=Λunif = 8MW |Λ=Λunif generations

Matilde Marcolli Early Universe Models Choice of initial conditions at unification: Compatibility with low energy values: experimental constraints Compatibility with geometric constraints at unification It is possible to modify boundary conditions to achieve both compatibilities Example: using maximal mixing conditions but modify parameters in the Majorana mass matrix and initial condition of Higgs parameter to satisfy geometric constraints

Matilde Marcolli Early Universe Models Cosmology timeline epoch: t ≤ 10−43 s after the (unification of forces with gravity, ) Grand Unification epoch: 10−43 s ≤ t ≤ 10−36 s (electroweak and strong forces unified; Higgs) : 10−36 s ≤ t ≤ 10−12 s (strong and electroweak forces separated) Inflationary epoch: possibly 10−36 s ≤ t ≤ 10−32 s - NCG SM preferred scale at unification; RGE running between unification and electroweak ⇒ info on inflationary epoch. - Remark: Cannot extrapolate to modern universe, nonperturbative effects in the spectral action and phase transitions in the RGE flow

Matilde Marcolli Early Universe Models Cosmological implications of the NCG SM Linde’s hypothesis (antigravity in the early universe) Primordial black holes and gravitational memory Running effective gravitational constant affecting gravitational waves Gravity balls Varying effective cosmological constant Higgs based slow-roll inflation Spontaneously arising Hoyle-Narlikar in EH backgrounds

Matilde Marcolli Early Universe Models Effective gravitational constant

2 κ0 3π Geff = = 2 8π 192f2Λ − 2f0c(Λ) Effective cosmological constant 1 γ = (192f Λ4 − 4f Λ2c(Λ) + f d(Λ)) 0 4π2 4 2 0

Matilde Marcolli Early Universe Models Conformal non-minimal coupling of Higgs and gravity 1 Z √ 1 Z √ R gd 4x − R |H|2 gd 4x 16πGeff 12 Conformal gravity −3f Z √ 0 C C µνρσ gd 4x 10π2 µνρσ

C µνρσ = Weyl curvature tensor (trace free part of Riemann tensor)

1 1 C = R − (g R −g R +g R )+ (g g −g g ) λµνκ λµνκ 2 λν µκ µν λκ µκ λν 6 λν µκ λκ µν

Matilde Marcolli Early Universe Models An example: Geff (Λew ) = G (at electroweak Geff is already√ modern universe Newton constant) 19 32 1/ G = 1.22086 × 10 GeV ⇒ f2 = 7.31647 × 10

96f Λ2 G −1(Λ) ∼ 2 eff 24π2 Term e/a lower order Dominant terms in the spectral action:  Z Z  2 1 √ 4 2 2√ 4 Λ 2 R gd x − µ˜0 |H| gd x 2˜κ0

2 κ˜ = Λκ andµ ˜ = µ /Λ, where µ2 ∼ 2f2Λ 0 0 0 0 0 f0

Matilde Marcolli Early Universe Models Detectable by gravitational waves: µν 1 µν 2 µν Einstein equations R − 2 g R = κ0T   2 −1 0 gµν = a(t) 0 δij + hij (x) trace and traceless part of hij ⇒ Friedmann equation

a˙ 2 1  a˙   κ˜2 −3 + 4 h˙ + 2h¨ = 0 T a 2 a Λ2 00

Matilde Marcolli Early Universe Models αt Λ(t) = 1/a(t) (f2 large) Inflationary epoch: a(t) ∼ e NCG model solutions:

2 3π T00 2αt 3α A −2αt h(t) = 2 e + t + e + B 192f2α 2 2α Ordinary cosmology: 4πGT 3α A ( 00 + )t + e−2αt + B α 2 2α Radiation dominated epoch: a(t) ∼ t1/2 NCG model solutions: 4π2T 3 h(t) = 00 t3 + B + A log(t) + log(t)2 288f2 8 Ordinary cosmology: 3 h(t) = 2πGT t2 + B + A log(t) + log(t)2 00 8

Matilde Marcolli Early Universe Models Same example, special case:

2 2 2˜κ µ˜ af0 p R ∼ 0 0 ∼ 1 and H ∼ af /π π2 0 Leaves conformally coupled matter and gravity Z √ 1 Z √ S = α C C µνρσ g d 4x + |DH|2 g d 4x c 0 µνρσ 2 Z Z 2 √ 4 4 √ 4 −ξ0 R |H| g d x + λ0 |H| g d x 1 Z √ + (G i G µνi + F α F µνα + B Bµν ) g d 4x. 4 µν µν µν A Hoyle-Narlikar type cosmology, normally suppressed by dominant Einstein–Hilbert term, arises when R ∼ 1 and H ∼ v, near see-saw scale.

Matilde Marcolli Early Universe Models Cosmological term controlled by additional parameter f4, vanishing condition: (4f Λ2c − f d) f = 2 0 4 192Λ4

Example: vanishing at unification γ0(Λunif ) = 0

5.0´1015 1.0´1016 1.5´1016 2.0´1016

-5.0´1092

-1.0´1093

-1.5´1093

-2.0´1093

-2.5´1093

Running of γ0(Λ): possible inflationary mechanism

Matilde Marcolli Early Universe Models The λ0-ansatz π2b(Λ ) λ | = λ(Λ ) unif , 0 Λ=Λunif unif 2 f0a (Λunif )

Run like λ(Λ) but change boundary condition to λ0|Λ=Λunif Run like π2b(Λ) λ0(Λ) = λ(Λ) 2 f0a (Λ) For most of our cosmological estimates no serious difference

Matilde Marcolli Early Universe Models The running of λ0(Λ) near the top see-saw scale

0.202

0.200

0.198

0.196

0.194

0.192

2´1014 4´1014 6´1014 8´1014 1´1015

Matilde Marcolli Early Universe Models Linde’s hypothesis antigravity in the early universe A.D. Linde, Gauge theories, time-dependence of the gravitational constant and antigravity in the early universe, Phys. Letters B, Vol.93 (1980) N.4, 394–396 Based on a conformal coupling 1 Z √ 1 Z √ R gd 4x − R φ2 gd 4x 16πG 12 giving an effective 4 G −1 = G −1 − πφ2 eff 3 In the NCG SM model two sources of negative gravity

Running of Geff (Λ) Conformal coupling to the Higgs field

Matilde Marcolli Early Universe Models −1 Example of effective Geff (Λ, f2) near the top see-saw scale

-6.26´1029 0.0001 -6.28´1029 -6.30´1029 -6.32´1029 0 0.00005

5´1014

0.0000 1´1015

Example: fixing Geff (Λunif ) = G gives a phase of negative gravity with conformal gravity becoming dominant near sign change of −1 12 Geff (Λ) at ∼ 10 GeV

Matilde Marcolli Early Universe Models Gravity balls G G = eff eff,H 4π 2 1 − 3 Geff |H| combines running of Geff with Linde mechanism Suppose f2 such that Geff (Λ) > 0  2 3  Geff,H < 0 for |H| > ,  4πGeff (Λ)  2 3  Geff,H > 0 for |H| < . 4πGeff (Λ) Unstable and stable equilibrium for H: 2 e(Λ) 2 f2Λ 2 µ 2 f − (2f Λ a(Λ) − f e(Λ))a(Λ) ` (Λ, f ) := 0 (Λ) = 0 a(Λ) = 2 0 H 2 2λ π2b(Λ) π2λ(Λ)b(Λ) 0 λ(Λ) 2 f0a (Λ)

(with λ0-ansatz) Negative gravity regime where 3 `H (Λ, f2) > 4πGeff (Λ, f2)

Matilde Marcolli Early Universe Models An example of transition to negative gravity

2.0´1030

1.5´1030

1.0´1030

5.0´1029

1´1014 2´1014 3´1014 4´1014 5´1014 6´1014 7´1014

Gravity balls: regions where |H|2 ∼ 0 unstable equilibrium (positive 2 gravity) surrounded by region with |H| ∼ `H (Λ, f2) stable (negative gravity): possible model of

Matilde Marcolli Early Universe Models Primordial black holes (Zeldovich–Novikov, 1967) I.D. Novikov, A.G. Polnarev, A.A. Starobinsky, Ya.B. Zeldovich, Primordial black holes, Astron. Astrophys. 80 (1979) 104–109 J.D. Barrow, Gravitational memory? Phys. Rev. D Vol.46 (1992) N.8 R3227, 4pp. Caused by: collapse of overdense regions, phase transitions in the early universe, cosmic loops and strings, inflationary reheating, etc

Gravitational memory: if gravity balls with different Geff,H primordial black holes can evolve with different Geff,H from surrounding space

Matilde Marcolli Early Universe Models Evaporation of PBHs by Hawking radiation

dM(t) ∼ −(G (t)M(t))−2 dt eff

−1 with Hawking temperature T = (8πGeff (t)M(t)) . In terms of energy: 1 M2 dM = dΛ 2 2 Λ Geff (Λ, f2)

With or without gravitational memory depending on Geff behavior

Evaporation of PBHs linked to γ-ray bursts

Matilde Marcolli Early Universe Models Higgs based slow-roll inflation dSHW A. De Simone, M.P. Hertzberg, F. Wilczek, Running inflation in the Standard Model, hep-ph/0812.4946v2 Minimal SM and non-minimal coupling of Higgs and gravity. Z 2√ 4 ξ0 R |H| gd x

Non-conformal coupling ξ0 6= 1/12, running of ξ0 √ Effective Higgs potential: inflation parameter ψ = ξ0κ0|H|

1.0

0.8

0.6

0.4

0.2

2 4 6 8 10 inflationary period ψ >> 1, end of inflation ψ ∼ 1, low energy regime ψ << 1 Matilde Marcolli Early Universe Models In the NCG SM have ξ0 = 1/12 but same Higgs based slow-roll inflation due to κ0 running (say κ0 > 0) s p π2 ψ(Λ) = ξ0(Λ)κ0(Λ)|H| = 2 |H| 96f2Λ − f0c(Λ)

E 2 Einstein metric gµν = f (H)gµν, for f (H) = 1 + ξ0κ0|H| Higgs potential λ |H|4 V (H) = 0 E 2 2 2 (1 + ξ0κ0|H| ) For ψ >> 1 approaches constant; usual quartic potential for ψ << 1

Matilde Marcolli Early Universe Models Conclusion: Various possible inflation scenarios in the very early universe from running of coefficients of the spectral action according to the relation to Yukawa parameters: phases and regions of negative gravity, variable gravitational and cosmological constants, inflation potential from nonmininal coupling of Higgs to gravity

Main problem: these effects depend on choice of initial conditions at unification (sensitive dependence) and several of these scenarios are ruled out when moving boundary conditions

Matilde Marcolli Early Universe Models Other cosmological aspects of the Spectral Action The problem of cosmic topology The Poisson summation formula Nonperturbative computation of the spectral action on 3-dimensional space forms Slow-roll inflation: potential, slow-roll coefficients, power spectra Slow-roll inflation from the nonperturbative spectral action and cosmic topology

Matilde Marcolli Early Universe Models